arXiv:1703.09645v1 [math.HO] 28 Mar 2017

Cognition of the circle in ancient India

S.G. Dani

Next only to the rectilinear figures such as triangles and rectangles, the circle

is the simplest geometrical figure that would have touched human life even in the

primitive stages, especially after the advent of the wheel. Apart from the everyday

secular aspects of life, the circle gained significance in ritual and spiritual respects.

Considerable understanding was acquired over a period in the ancient times, con-

cerning various geometric features of the circle. Progress in understanding the

circle may be readily correlated with progress of human civilization in general.

Our overall knowledge of history across ancient cultures has many limitations, in

terms of source material and means of interpretation. Nevertheless, in the Indian

context we are endowed with information from various sources such as Sulvasutras,

the Jaina compositions, works from the mathematical astronomy tradition start-

ing with Aryabhat.a, and finally the Kerala school of mathematics, from different

periods in history, that give an interesting perspective on how ideas developed on

the issue.

The Sulvasutra period

The Sulvasutras are compositions concerned with construction of altars (Vedi)

and fire platforms (citi) for the Vedic rituals.1 Apart from elaborate instruc-

tion on laying of bricks (of simple rectilinear shapes, such as squares, triangles

etc.) to achieve approximations to various elaborate shapes such as falcons, tor-

toise, wheel, etc., the compositions also include enunciation of various geometric

principles, geometric constructions with practical or theoretical import, etc., thus

providing us a glimpse of the mathematical knowledge at that time. There were

many Sulvasutras, of which Baudhayana, Apastamba, Manava and Katyayana

Sulvasutras are especially noted for their significance from a mathematical point

of view. The period of the Sulvasutras is somewhat uncertain, as there are no in-

ternal clues in the compositions other than their style and language, but there now

seems to be a general consensus among scholars that they belong to the period

from about 800 BCE to 200 BCE, Baudhayana being the oldest and Katyayana

the latest. For further general details the reader is referred to [18], [12], [1], and

1Performance of yajnas, fire rituals, in pursuance of material and/or spiritual benefits is oneof the dominant features of the Vedic civilisation. It involved both the royalty as well as laityfrom the priestly class of the time. There are detailed prescriptions, about specific yajnas to beperformed for various objectives, as well as the procedures to be followed.

1

arX

iv:1

703.

0964

5v1

[m

ath.

HO

] 2

8 M

ar 2

017

the references cited there - here we shall focus on the specific theme at hand,

concerning the circle.

One of the simplest questions that one can think of about the circle is the ratio

of its circumference to the diameter. As in other ancient cultures, in ancient India

also this ratio was believed to be 3. In the context of the Vedic tradition this is

reflected in an indirect reference in the Baudhayana Sulvasutra in the statement

“The pits for the sacrificial posts are 1 pada in diameter, 3 padas in circumference.”

(Baudhayana Sulvasutra 4.15, see [18]); pada, which literally means foot, was a

measure of length equivalent to about 28 cm. The second part of the statement is

evidently meant as an elaboration/clarification of the first part, but provides us

a clue that they considered the circumference to be 3 times the diameter.

The choice of the value 3 for the ratio would today seem quite surprising, as

one would expect that many everyday experiences could have suggested the value

to be a little more. The following seems to me to be a plausible explanation for

this (which does not seem to have appeared in literature before): the idea of the

ratio being 3 dates back to the time when mankind was yet to think in terms of

fractions (except perhaps for “half”, which may have meant a substantial portion

of the whole, rather than its precise value as we understand it) and developed into

a belief (perhaps linked with religion). The ratio was assigned the value 3 in the

sense that it is not, say 2 or 4, or even “three hand a half”. The belief, once it was

rooted deeply, was not reviewed for a long time, even after fractions became part

of human thought process. While our encounter with the circle, especially in the

context of wheels, is over 5000 years old, fractions seem to have appeared on the

scene in a serious way, in Indian as well as Egyptian cultures, substantially later,

possibly only in the first millennium BCE. The difference between the actual ratio

and 3 is small enough not to come in to serious conflict with everyday experience

to warrant doubting an accepted proposition, which furthermore may have the

backing of religious authority, and an appeal on account of universal significance

associated with the number three. Also, while for a first-time determination of an

entity one typically avails of prevalent techniques of any given time, a belief often

remains untested until coming in conflict with another idea or experience.

The Manava Sulvasutra however breaks out of the mould, and we encounter

the following:

vis.kambhah.pancabhagasca vis.kambhastrigun.asca yah. |sa man.d.alapariks.epo na valamatiricyate ||

(Manava Sulvasutra 10.3.2.13)

A fifth of the diameter and three times the diameter, that is the cir-

2

cumference of the circle, not even a hair-breadth remains.

Apparently over the years it was recognised that the ratio is indeed a little

larger than 3. Manava seems to have taken a leaf out of this and came up with a

better estimate. The exultation over it is striking!

Unlike the circumference, the area of the circle is seen to have been of direct

interest to the authors of the Sulvasutras. There is no indication in the Sulvasutras

of their being aware of the ratio of the circumference to the diameter being the

same as the ratio of the area to the square of the radius; no occasion seems to

have presented itself that would inspire a comparison of the two ratios. The issue

of area, which was involved in the construction of altars, is treated independently.

There were citis (fire platforms) constructed in the shape of a chariot wheel, a

circular trough etc. with stipulated areas, which motivated the issue of how to

transform a square into a circle having the same area.

Transforming a square into a circle

Baudhayana describes a procedure to produce a circle with the same area as a

given square, which goes as follows: take a string with length half the diagonal of

the square, and stretch from the centre across a side of the square, viz. PS as in

Figure 1, and draw the circle including a third of the extra part stretching outside

the square, viz. PR as in the figure with QR=13

PS, as the radius.

Circling the square, Baudhayana Sulvasutra

For a square with side 2a this radius works out to be a + 13(√

2 − 1)a =

(2 +√

2)a/3. For the unit square the area of this circle works out to 1.01725...,

3

about 1.7% more than the correct value 1. If one is to compute the value of π with

(2 +√

2)a/3 as the radius of the circle corresponding to a square with side 2a,

it works out to be 3.0883 . . . , in place of 3.1415 . . . . It should be borne in mind

that what they had was a procedure for producing the circle and not a numerical

value for π; the latter had not emerged as a concept, and they were not trying

to compute such a ratio. The comparison, here and in similar contexts below, is

only for facilitation in overall comprehension of the relative values.

The Apastamba Sulvasutra gives the same construction for the circle, as

Baudhayana. Manava Sulvasutra is seen to provide another construction for the

circle with the area of the given square. The following interpretation of a verse in

Manava Sulvasutra was introduced in [1] by this author. For convenience I shall

discuss the verse and background around it, after first describing the procedure,

according to the interpretation. The steps involved are illustrated by Figure 2.

Circling the square, Manava Sulvasutra

Draw the lines dividing the square into 3 equal strips. Produce one

of these lines to meet the circle passing through the vertices of the

square. On the segment of the line that is outside the square and

inside the circle, viz. QS as in Figure 2, take the point at a distance15th of the length of the segment, from the square, viz. the point R

in the figure, with QR=15QS. The circle with PR as the radius, where

P is the centre of the square, is given as the desired circle, with area

equal to that of the original square.

4

For a square of side length 2 the length of the segment between the square and

the outer circle is seen to be

(√17

3− 1

), so the radius r of the circle is given by

r2 =

{1 +

1

5

(√17

3− 1

)}2

+1

9.

For the unit square the area of this circle works out to 0.9946 . . . , a substantially

more accurate value compared to the earlier one, the error involved being just

about 12% (which is now on the other side). The value of π in this case works out

to 3.1583 . . . .

The verse in question, from Manava Sulvasutra is:

caturasram. navadha kuryaddhanuh.kot.yastridhatridha|utsedhatpancamam. lumpetpurıs.en.eha tavatsamam. ||

(Manava Sulvasutra 11.15)

There seems to be considerable confusion about, and discomfort with, this

verse in literature. In [18] it is suggested that possibly “squares are drawn without

any mathematical significance”. In [12] there is an interpretation, the conclusion

of which manifestly wrong. There is another interpretation in [7], concluding

the value of π according to the sutra to be 258

, but it may be seen that in the

interpretation the first line of the verse plays no role at all, while that of the

second is quite ad hoc. Appropriate transliteration seems to be at the heart of

the issue.

I propose the following translation:

Divide the square into nine parts, (by) dividing the sides into three

(equal) parts each. Mark a fifth of the part jutting out (of the square)

and cover (the corresponding circle with centre at the origin) with

loose earth.

The meaning of this (according to my interpretation) is described above, with

the help of Figure 2. It would be out of place to go into the linguistic details with

regard to the interpretation. I shall instead focus on highlighting two reasons for

which the present interpretation ought to be appropriate. Firstly, as noted above,

it leads to a significantly improved result; this could not be a mere coincidence.

Secondly if one is to try to read their mind on how they might have attempted

5

to remedy a perceived discrepancy in a known result, the construction seems to

arise as a natural development: In the first place, it is reasonable to suppose in

this respect that over a period it had been realised that the circle produced by

the Baudhayana construction was slightly larger than it should be. Since taking

a point on the bisector of the square along a side did not seem to work, they

chose to consider trisectors of square. So far there is no divergence in various

interpretations. The crucial, and distinctive point in the above interpretation is

that they picked a point on the trisecting line, which is actually natural in the

context of the comparison with the Baudhayana construction, but does not seem

to have been taken note of by the earlier authors. Furthermore in analogy with

the earlier construction a point had to be picked up on the segment of the trisector

jutting out from the square. In the earlier construction one third of the jutting

out part was added to get the radius of the desired circle, and it may be noted

here that though the circle through the vertices of the square finds no mention in

the verse, it would be lurking in the their minds, in the context of the Baudhayana

construction. The fraction 15th of the extra part is then likely to have been based

on an ad hoc observation that the resulting line segment for the radius is slightly

smaller than the Baudhayana construction, as was desired.

Clearly, the Manava construction is the result of keen attempts to improve

upon the original Baudhayana construction, through refinement of the overall

scheme. How the specific details were conceived and how, and to what extent, it

was confirmed to be more accurate, remains unclear.

It may be mentioned here that there were also other constructions adopted,

in the broader Vedic community; while indeed the Vedic civilization shared a

certain common body of knowledge, there are many variations in the individual

Sulvasutras adopted by different sub-communities. A lesser known Sulvasutra by

the name Maitrayan. ıya, which is akin to Manava Sulvasutra (in that it belongs

to the same samhita), gives a construction for circling the square which involves

taking the radius of the desired circle to be 916

times the side of the square;

see [9]. For a unit square the area of the resulting circle turns out to be 0.9940 . . . ,

comparable to the one above, with 3.1604 . . . as the corresponding value of π. It

may be recalled here that the Egyptians took the area of a circle of radius r to be

(16r9

)2, to which the above value, for the reverse process, corresponds exactly.

Squaring the circle

The converse problem of “squaring the circle”, viz. finding a square with the

same area as a given circle2 is also considered in the Sulvasutras. Baudhayana

2This should not be confused with the problem in Greek geometry of finding a square with

6

gives the following expression for the ratio of the side of the square to the diameter

of the circle (the original description is in words):

7

8+

1

8× 29− 1

8× 29× 6+

1

8× 29× 6× 8. (1)

For the circle with unit radius, the area according to this works out to 3.0883...,

a little more than 98.3% of the actual value. It may be noted that in this case

also the error is about 1.7 percent, in the opposite direction. It could not be a

coincidence (as has been noted also by earlier authors - see [19], [17]), that the

errors in the two prescriptions, corresponding to mutually opposite operations,

while substantial, are quite matching in their order and opposite in the orientation.

It suggests that for want of a geometrical procedure in the reverse direction (unlike

for transforming a square into a circle) they obtained it through inversion of the

previous ratio, in some way which is not entirely clear so far (see below). To get

the inverse of (2 +√

2)/3 they sought out a value of√

2, as a familiar fraction.

The square root of 2

Three of the four Sulvasutras, Baudhayana, Apastamba and Katyayana, give

the following expression for√

2 (in words):

1 +1

3+

1

3× 4− 1

3× 4× 34. (2)

In decimal expansion the value of the expression is 1.4142157 . . . . This is remark-

ably close to the actual value 1.4142136 . . . , and this fact has been a subject of

much laudatory comment in literature. It may be recalled in this context that

Babylonians also had a value, about a thousand years earlier, describing√

2 in

the sexagesimal system, which works out to 1.4142129... (see [4], for instance).

Various aspects including the presentation of the number as above and the sub-

stantial relative difference of the values (including the side of the error), rule out

any organic link between the values. There has been considerable speculation and

discussion on how the Sulvasutra value of√

2 may have been arrived at; we shall

however not digress to these details here (see for instance [1] and other references

cited there).

precisely the same area through a ruler and compass construction. The context of the sutrakaraswas entirely different, and their objective would have been only to find a square with the areaof the circle, within the levels of accuracy they were used to, or desired. They may have likedto find a geometric construction, like by ruler and compass, but that was not the thrust. Theproblem in their perspective involved finding such a square by whatever available means.

7

As noted above the motivation for finding a value for√

2 would have come

from the problem of computing the inverse of (2 +√

2)/3, viz. for obtaining

the formula (1). This numerical value of√

2 is not involved elsewhere in the

Sulvasutras. In other contexts they are seen to use only the geometric form of√2 as the diagonal of the unit square, which in fact went by the special name

dvikaran. ı.

How the inversion would have been effected, using the value of√

2 as above,

has been discussed by Thibaut [19] and also other later authors. The older expla-

nations, however, presuppose considerable dexterity on the part of the sutrakaras

in dealing with fractions, for which there is no corroborative evidence, and are

thus unsatisfactory. In a recent paper Kichenassamy [11] has proposed a resolu-

tion of the issue which is more in tune with the Sulvasutras ethos; the paper also

discusses at length the inadequacies of the earlier arguments.

It would also be worthwhile to note here another Sulvasutra construction which

relates in a way to properties of the circle. Baudhayana Sulvasutra describes a

construction of a square which involves drawing a perpendicular to a given line,

say L, at a point P on L, by drawing circles with centres at points on either side

of P on L at equal distance, with radii larger than the distance, and joining the

points of intersection of the two circles (in very much the same way as taught in

schools today). Underlying the construction is the realisation that the line joining

the two points of intersection of two circles meets the line joining their centres

orthogonally; though the construction involves the principle for circles of equal

radius, it seems reasonable to assume that they were aware of this “orthogonality

principle” in that generality. In most constructions requiring perpendiculars, they

were however produced using the converse of the Pythagoras theorem3, rather than

the construction as above (implemented in a certain way, the former turns out to

be simpler than the latter; see [1]).

Let me conclude this section on the Sulvasutras with the following com-

ment. There has been a tendency with regard to Sulvasutras to assume that

the sutrakaras lay great store on accuracy. While the value of√

2 does seem like

an example of this, a careful reading of the Sulvasutras shows that high degree

of accuracy was not seen as a primary objective. In many contexts, alternate

values or constructions are described, which are of a crude variety, alongside some

3The theorem named after Pythagoras has been known in India at least since the time of theearliest Baudhayana Sulvasutra (ca. 800 BCE), where an explicit statement of it is found. Theconverse of the theorem, namely that a triangle in which the square of one of the sides equalsthe sum of squares of the other two sides, was used extensively for producing perpendiculars(see [1] for details).

8

relatively accurate ones, which shows that in their overall conception, the benefits

meant to accrue from the ritual performances would not be seriously affected if

approximate procedures were adopted. Where accuracy was pursued, it seems to

be the result of keen academic enquiry, rather than an imperative arising from

practical issues of the time, or the philosophical framework involved. On the other

hand a supposition as above does them a disservice in the context of the less accu-

rate values such as in circling a square. Mathematics of ancient cultures needs to

be understood and appreciated in their specific context, and not judged through

generalised abstract tests. The issue of circling the square arose for instance from

the desire of having a fire platform with the same area, that would not bring with

it an intrinsic demand for high degree of accuracy, and it is incorrect to wonder

why their value of the area is not accurate - it was not meant to be.

The Jaina tradition

Apart from the Vedic religion (if one may call it that) Jainism and Buddhism

flourished during the first millennium BCE (and later during certain periods).

There was a long tradition among the Jainas of engagement with mathematics, as

is evidenced from various compositions that have come down to us. As for Bud-

dhism, though certain constructions involved in Buddhist pursuit, called Mandalas

involve intricate designs which seem mathematically significant, no textual com-

position involving mathematical concepts has come down to us.

The motivation of the Jainas for mathematics did not came, per se, from any

rituals, which they indeed abhorred, but from contemplation of the cosmos, of

which they had evolved an elaborate and unique conception of their own. In the

Jaina cosmography the world is supposed to be an infinite flat plane, with concen-

tric annular regions surrounding an innermost circular region with a diameter of

100000 yojanas4, known as the Jambudvıpa (island of Jambu, that corresponded to

the Earth), and the annular regions alternately consist of water and land, and the

width of each successive ring being twice that of the previous one. The geometry

of the circle played an important role in the overall discourse, even when the schol-

ars engaged in it were more of philosophers than mathematicians. Unfortunately,

many historical and chronological details of the Jaina tradition are uncertain (even

more so than the Hindu tradition) and have been a subject of speculation. Defini-

tive references to the properties of the circle known in the Jaina tradition can be

found in the work in fourth or fifth century ([14], page 59). It has however been

mentioned by Datta in [2] that they are found in Tattvarthadhigama-sutra-bhas.ya,

4yojana was a measure of length, of the order of 15 to 20 kilometres, with local variations.

9

a philosophical work of Umasvati, who is supposed to have lived around 150 BCE

according to the Svetambara tradition and in the second century CE according

to the Digambara tradition5. Datta [2] suggests that Umasvati was probably not

the discoverer of the formulae and they would have been known centuries before

him, and discusses some evidence in this respect. Saraswati Amma ([15], page 63)

attributes the basic formulae to Suryaprajnapti, a composition which is believed

to be from the fifth century BCE.

In the Jaina tradition the departure from old belief of 3 as the ratio of the

circumference to the diameter is quite pronounced; Suryaprajnapti records the

then traditional value 3 for it, and discards it in favour of√

10. The Jainas also

knew the ratio of the area of the circle to the square of the radius to be the

same number as the ratio of the circumference to the diameter. In fact they had

the formula directly relating the area to the circumference, that the former is a

fourth of the product of the circumference and the diameter of the circle, which

in particular readily implies the equality of the two ratios as above. Incidentally,√10 which is about 3.16227 . . . , may be seen to be a better approximation for π

compared to the Baudhayana construction, involving an error of only about 23

per

cent.

The value was very convenient to the Jaina theologian mathematicians, in their

computations. For example, in Jambudvıpa prajnapti the value the circumference

of Jambudvıpa, a circle of diameter 100, 000 yojana, is computed, with√

10 as

the value for the ratio of the circumference to the diameter, by computing the

square-root of 1011.

This value for π was used for over a thousand years, even after better values

were known; indeed so routine was its use in the Jaina texts that it is often known

as the Jaina value for π. The value was also adopted in Pancasiddhantika, in the

Siddhanta tradition sometime during 1st to 6th century, and by Brahmagupta in

the 7th century.

There has been some speculation on the origin of√

10 as a value for π. One

explanation, attributed to Hunrath, goes as follows ([15], page 65): The square of

the side of a regular 12-sided polygon inscribed in a circle of unit radius is seen

to be (1 −√32

)2 + (12)2 and with the choice of 5

3as an approximation for

√3 it

works out to√1012

; thus the perimeter of a regular 12-gon is about√

10 times its

principal diagonals, but a little less. This may have inspired the formula for the

circumference of the circle. The explanation however involves many assumptions

5Svetambara and Digambara are two ancient branches of Jainism, with certain differencesboth in terms of their philosophy as also practices in everyday life.

10

about which there is little evidence. An interested reader may also consult [6] for

various other explanations.

Vırasena, a Jaina mathematician from the 8th century states:

vyasam s.od.asagun.itam. s.od.asa sahitam. triruparupairbhaktam. |vyasam trigun.itam. suks.madapi tadbhavet suks.mam. |

(S. at.khan.d. agama, Vol. IV. page 42)

A routine translation of this would be as follows (slightly simplified from [17]):

Sixteen times the diameter, together with 16, divided by 113 and thrice

the diameter is a very fine value (of the circumference).

There is something strange about the formula (with the interpretation as

above), that it prescribes “together with 16” - surely it was known to the au-

thor that the circumference is proportional to the diameter and that adding 16,

independent of the diameter, would not be consistent with this. It seems reason-

able however to suppose that the author meant 3 + 16113

= 355113

to be the factor by

which to multiply the diameter to get the circumference, which is indeed a good

approximation, as the author stresses with the phrase “sukshmadapi sukshamam”

(finest of the fine !).6 In China this approximation for π was given by Chong-Zhi

(429-500). Its value is 3.1415929... in place of 3.1415926..., accurate to 6 decimals.

In Trilokasara, which is another account of the Jaina scholarship, composed

by Nemicandra, who lived around 980 CE, also one finds another value for the

ratio π, apart from√

10: it is the value (169

)2, that we saw from the Maitrayanıya

Sulvasutra (shared also with the Egyptians). This may suggest a relation with

the Hindu tradition, but the time gap is rather intriguing.

Apart from the circle as a whole, the Jaina mathematicians were also interested

in the interrelation between the arcs of a circle and the corresponding chords. This

is related to their conception of the geography of Jambudvıpa, including various

6There does not seem to be much of scope for attributing the issue to corruption in the courseof transmission at some level; there are however issues of grammar and interpretation involved,and this accepted translation may be flawed; it is possible, for instance, that s.od. asa sahitam.(”together with sixteen”), which is the culprit, has the role of emphasising that while dividingby 113, one is to divide the previous product, which involved 16 - thus “together with 16” isnot about adding 16, but reiterating that the following division by 113 is to be subjected to theoutput together with the earlier 16. While this indeed does seem odd in what is intended tobe a formula, it need not be ruled out, given that the direct interpretation is odd anyway, andthe author obviously would not have meant it. In part, the somewhat curious presentation mayhave been been the result of needs of versification.

11

regions, mountains etc. Umasvati notes various relations between the length c of

a chord, the height h of the corresponding “arrow” (viz. the segment joining the

midpoint of the chord to the midpoint of the arc) and the diameter d of the circle.

One of the relations noted is

c =√

4h(d− h);

various other forms which are equivalent to this one algebraically, from a modern

point of view, are also presented. An interested reader may consult [17], [2] and

also [8] for further discussion on this issue; the last two references have some

details also of analogous formulae from other ancient cultures.

There is also an interesting formula for the length of the minor arc (the smaller

of the two arcs cut out by the chord), say a, as

a =√

6h2 + c2

(with notation as above). As can be seen, such a relation does not actually hold

exactly. It may be noted that in the special case when the chord is a diameter,

so that the arc is a semicircle, the equation corresponds to the ratio of the length

of the semicircle to the radius being√

10; so the relation holds with their value

for π. As we go to small arcs however the assertion goes quite off the mark.

Surprisingly however, the formula continued to be part of Jaina literature all the

way, including the famous mathematical work Gan. ita sara sangraha of Mahavıra

in 850 (Ch. VII, verse 7312; see [14], page 469). The formula also appears in

Trilokasara of Nemicandra (see [3]), who was mentioned above.

Given a chord of a circle, apart from the length of the arc segment one may

also ask about the area cut out by the chord (with the minor arc). Gan. ita sara

sangraha gives the value of the area to be 14

√10ch, where c is the length of the

chord and h is the height over the chord (length of the arrow). The formula is

also found in Trilokasara of Nemicandra. This formula also holds strictly only for

a semicircular segment with π in place of√

10 but diverges from the actual value

for smaller arcs.

A different formula for the area of the segment cut out by the chord is given

in Trisatika of Sridhara (ca. 750)7, and also quoted in some later works, including

Bhaskara (see below for more about him). It may be worth mentioning here that

Sridhara does not quite seem to fit in the astronomer mathematicians tradition -

7Though until some time there had been an argument over when he lived and his background,there is now a general consensus that Sridhara is from the 8th century, and was Jaina, at leastduring the time of his writings - his mathematical work is seen to be consistent with this.

12

his known works deal exclusively with mathematics, and he is well-known for his

procedure for solving a quadratic equation. According to Sridhara the area A of

the segment between a chord and the corresponding arc is given by

A =

√10

3

(h(c+ h)

2

);

Clearly√

10 here is meant to be for the ratio π.

It would seem that many formulae for arc segments cut out by chords were

written down by extrapolating relations that were noted for the case of the semi-

circle to a general arc segment; if they had some (heuristic) reasoning for it, it is

not found recorded. From a historical point of view this highlights the difficulties

faced by the ancient mathematicians in grasping the lengths of arcs and the areas

bounded by them, and their endeavour to get around the difficulties, before the

ideas of trigonometry, and then calculus emerged.

Aryabhat.a and the astronomical tradition

Aryabhat.a, born in 476 CE (as has been indicated by the author in his work

Aryabhat.ıya), was the pioneer of what is termed as the siddhanta tradition, of as-

tronomer mathematicians in India that flourished for almost eight hundred years,

until Bhaskaracharya in the 12th century, and even beyond, and in turn led to

the Kerala school of mathematics. While the tradition has some manifest linkages

with the older Hellenistic mathematical astronomy, after the early influences it

seems to have charted a course of its own. Many new mathematical ideas were

developed, both in response to the theoretical demands in the study of astron-

omy, and also in pursuit of pure mathematical thought. In particular a deeper

understanding of the circle evolved, both in terms of geometry and trigonometry.

In his work Aryabhat.ıya we find the following:

Caturadhikam satamas.tagun.am dvas.astistatha sahasran.am |ayutadvayavis.kambhasyasanno vr.ttaparin.ahah ||

(Gan. itapada 10, in Aryabhat.ıya)

The circumference of a circle with diameter twenty thousand is ap-

proximately a hundred and four times eight, and sixty-two thousand

[viz. 62832].

This gives the value of π as approximately 3.1416, which indeed coincides with the

correct value of π truncated at 4 decimal places. It may be recalled that in Greek

13

astronomy, Ptolemy had the value, in sexagesimal expression, which corresponds

to 3.14166.... There is no direct information on how Aryabhata arrived at the

value. One may anticipate that, like in similar instances in other cultures, the

value was obtained through repeated application of the formula

S2n =

√√√√S2n

4+

(1−

√4r2 − S2

n

4

)2

,

where Sn is the side of the regular n-gon inscribed in a circle of unit radius. The

formula follows from the “Pythagoras theorem”, which, as noted earlier, had been

known in India since the Baudhayana Sulvasutra (8th century BCE) and is also

stated in Aryabhatıya (in Gan. itapada - 17). It is suggested by Ganesa, a sixteenth

century commentator of Aryabhat.ıya, that an inscribed polygon with 384 sides

was used as an approximation for the circle, and the above formula was used,

starting with a hexagon (for which the side coincides with the radius of the circle),

until reaching the polygon with the number of sides 384 = 6 × 26. The choice

of 20,000 as the measure for diameter is readily seen to facilitate computation of

square-roots in integral values; the values would have been rounded, up or down,

to integer values at various stages of application of the above formula, and the

square root computed using the well-known procedure for the purpose, that is

attributed to Aryabhat.a. It may be noted that the value of π as above is slightly

greater than the actual value, despite its representing the perimeter of an inscribed

regular polygon, due to rounding up at some stages.

In Aryabhatiya we also have the trigonometric sine functions.8 Aryabhat.ıya

(499 CE) provides a sine table, in a verse, for angles upto 90◦ that are multiples

of 3◦45′ (24th part of the right angle): taking the circle whose circumference is

21600 (equal to the total measure of the circumference in minutes), the differences

between the values of half-chords corresponding to angles that are successive mul-

tiples of 3◦45′ are recounted sequentially; the radius of the circle, which features

as the total of the differences recounted, is 3438. A similar table also appears in

Panca-siddhantika an older composition from the early centuries of CE in which

the value of the radius involved is 120. Once such tables were available, the lengths

of circular arcs could be calculated using the sine table (for the specific values),

without recourse to any special formula as in Jaina mathematics. There were also

interpolation methods for dealing with intermediate angles.

8While the Greeks did trigonometry with chords, it was in India that the trigonometry interms of the half-chords originated.

14

Apart from the sine tables there was also a curious approximate formula for

the sine function in vogue in the Siddhanta tradition. It is generally attributed to

Bhaskara I (7 th century CE), being part of his Mahabhaskarıya, but is also

found independently in the contemporaneous work Brahmasput.a siddhanta of

Brahmagupta. In the modern notation the formula may be stated as

sin θ =4θ(180− θ)

40500− θ(180− θ),

where θ is the angle measured in degrees. The formula is seen to be remarkably

accurate, involving an error of less than 1%, except for very small angles. It is

unclear how such a formula was derived. (see [5] and [20] for further details in

this respect).

Knowledge of various properties of the circle and trigonometry gradually be-

came crucial part of learning in the Siddhanta tradition, being a prerequisite for

pursuing mathematical astronomy. The tradition sustained itself, though perhaps

somewhat feebly during certain periods than others, and individual exponents

made fresh contributions to knowledge, apart from carrying forward the body

of knowledge that was getting built. We shall not go into the finer historical

details in this respect here. Bhaskara II, from the 12th century, (also known

as Bhaskaracarya, Bhaskara the teacher) is considered the last major exponent

from the tradition. Apart from mastering the knowledge flowing in the tradition,

Bhaskara made substantial contributions of his own in various respects. By his

time, the attendant mathematics, especially arithmetic and geometry, that went

with mathematical astronomy, had acquired a wider appeal, and applicability, in

the society. Bhaskaracarya composed a comprehensive work, Siddhanta Siroman. i

which, in the tradition of Siddhanta works, had a chapter devoted the mathe-

matical topics as above, called Lılavatı. The latter however acquired a life of

its own, and a reputation as a mathematical work, with large number of copies

being produced. It served as a textbook of mathematics for several centuries, in

a large part of India. Specifically with regard to the circle I will only recall the

following (approximate) formula from Lılavati for the length of an arc of a circle;

the formula itself may be seen to be related to Bhaskara I’s formula for the sine

function, when expressed in radians:

a =p

2−

√p2

4− 5p2c

4(c+ 4d)=p

2

(1−

√1− 5c

c+ 4d

),

where p denotes the circumference (perimeter) of the circle, and the other notation

is as above, namely a is the length of the arc, c is the length of the chord, and d

15

is the diameter of the circle; the first expression as above is akin to the way it is

given in the original verse and the second is a simplification. A more integral view

is seen to have evolved with regard to geometry of the circle and trigonometry.

The Kerala School

We conclude this article with a few observations on the Kerala school in the

context of the above theme. The school originated with the work of Madhava

in the second half of the 14th century, and flourished, as a teacher-student con-

tinuity, with multiple names involved during some periods, for about 250 years.

They took remarkable strides towards calculus, introducing techniques involving

infinitesimals, and in particular had obtained Gregory-Leibnitz series for the arc-

tan function and the Newton series for the sine function (over two centuries before

their European counterparts). We shall not go into a detailed discussion on the

mathematics from the Kerala school, which has been a subject of much study in

recent years. The interested reader is referred to [10], [14], and [16].

Determining accurate values for π, which is something that concerns our theme

here, seems to have been a passion for the school. In particular the following

remarkably close approximation to π is credited to Madhava, by Sankara Variar

(1556), in Kriyakramakarı (cf. [14]): the measure of the circumference in a circle

of diameter 900, 000, 000, 000 is 2, 827, 433, 388, 233. Thus

π =2, 827, 433, 388, 233

900, 000, 000, 000= 3.141592653592 . . . ,

in place of 3.141592653589 . . . , accurate to 11 decimals, when rounded. As the

series expansion

Circumference = 4 diameter (1− 13

+ 15

+ . . . )

that they had obtained converges very slowly, and hence not useful in getting

good approximations for π. Madhava had introduced an ingenious device to get

over this difficulty, called antya samskara, “the end correction”. With Sn as the

sum of the series truncated at the nth term, he introduced sequences an such that

the sequence Sn + an converges faster. The third, the final one that was recorded,

produces the sequence

Sn + (−1)n−1n2 + 1

4n3 + 5n.

The 50 th term of this is accurate to 11 decimals.

16

References

[1] S.G. Dani, Geometry in the Sulvasutras, in Studies in History of Mathematics,

Proceedings of Chennai Seminar, Ed. C.S. Seshadri, Hindustan Book Agency, New

Delhi, 2010.

[2] Bibhutibhushan Datta, The Jaina School of Mathematics, Bull. Cal. Math. Soc. 21

(1929), 115 -145.

[3] Bibhutibhushan Datta, Mathematics of Nemicandra, The Jaina Antiquary, 1

(1935),

[4] D. Fowler and E. Robson, Square Root Approximations in Old Babylonian Math-

ematics: YBC 7289 in Context, Historia Math. 25 (1998) 366 - 378,

[5] R.C. Gupta, Bhaskara I’s approximation to sine, Indian J. History Sci. 2 (1967),

121 - 136.

[6] R.C. Gupta, Madhavacandra’s and other octagonal derivations of the Jaina value

π =√

10, Indian J. Hist. Sci. 21 (1986), no. 2, 131 - 139.

[7] R.C. Gupta, New Indian values of π from the Manava sulba sutra, Centaurus 31

(1988), no. 2, 114 - 125.

[8] R.C. Gupta, Area of a bow-figure in India, Studies in the history of the exact

sciences in honour of David Pingree, 517 - 532, Islam. Philos. Theol. Sci., LIV,

Brill, Leiden, 2004.

[9] R.C. Gupta, Sulvasutras: earliest studies and a newly published manual, Indian J.

Hist. Sci. 41 (2006), 317 - 320.

[10] George Gheverghese Joseph, A passage to infinity, Medieval Indian mathematics

from Kerala and its impact, Sage Publications, Los Angeles, CA, USA, 2009.

[11] S. Kichenassamy, Baudhayana’s rule for the quadrature of the circle, Historia Math-

ematica 33 (2006), 149-183.

[12] Raghunath P. Kulkarni, Char Sulbasutra (in Hindi), Maharshi Sandipani Rashtriya

Vedavidya Pratishthana, Ujjain, 2000.

[13] Padmavathamma, Srı Mahavıracarya’s Gan. itasarasangraha, Publ.: Sri

Siddhantakrıithi Granthamala, Sri Hombuja Jain Math, Hombuja, Shimoga

Dist., Karnataka - 577436, India, 2000.

[14] Kim Plofker, Mathematics in India : 500 BCE - 1800 CE, Princeton University

Press, NJ, USA, 2008.

17

[15] T.A. Saraswati Amma, Geometry in Ancient and Medieval India, Motilal Banar-

sidas, Delhi, 1979.

[16] K. Ramasubramanian and M.D. Srinivas, Development of calculus in India, Studies

in the history of Indian mathematics, 201 - 286, Cult. Hist. Math., 5, Hindustan

Book Agency, New Delhi, 2010, 201-286.

[17] A. Seidenberg, The ritual origin of geometry, Archive for History of Exact Sciences,

1 (1962), 488 - 527.

[18] S.N. Sen and A.K. Bag, The Sulvasutras of Baudhayana, Apastamba, Katyayana,

and Manava, Indian National Science Academy, 1983.

[19] G. Thibaut, The Sulvasutras, The Journal, Asiatic Society of Bengal, Part I, 1875,

Printed by C.B. Lewis, Baptist Mission Press, Calcutta, 1875.

[20] Glen Van Brummelen, The mathematics of the heavens and the earth; The early

history of trigonometry, Princeton University Press, Princeton, NJ, USA, 2009.

Note: The papers of R.C. Gupta cited here are also available in the compilation

of Gan. itananda, edited by K. Ramasubramanian, Published by the Indian Society

for History of Mathematics (ISHM), 2015.

Department of Mathematics

Indian Institute of Technology

Powai, Mumbai, 400005

E-mail: [email protected]

18